Graphs of Trig Equations, Part 3

I know it’s taking a while before I use maths to model a mass on a spring, but that will only make sense by fully describing the graphs of sine equations. Hopefully, this development is interesting in its own right.

Now if you were to plot the daylight length at a certain latitude against days, and if you plotted for a full year, you would see a shape that looks amazingly like the sine graph I showed you in my last post. Except at the equator, the length of a day gets longer in the summer and shorter in the winter. Without actual taking a year to collect the data, I’ve plotted the daylight length in Melbourne Australia against days using the equation

\[
{L}\hspace{0.33em}{=}\hspace{0.33em}{2}{.}{63}\sin\left({\frac{360t}{365}}\right)\hspace{0.33em}{+}\hspace{0.33em}{12}{.}{165}
\]

where L is the length of daylight in hours and t is the number of days after 22 September of any year. Why I chose 22 September is an interesting topic which I may eventually discuss, but it has to do with what are called equinoxes. The plot is below and is almost exactly the plot created if I actually measured the day length each day and plotted these for a year:

Now the shape of this curve is a sine wave but you can see several differences from the standard sine wave explored in my last post:

  1. The amplitude is 2.63 instead of 1
  2. The wavelength is 365 days instead of 360 degrees
  3. The wave is centered at 12.165 instead of the x-axis
  4. We are evaluating the sine of time instead of degrees.

Let me explain these differences.

  1. Amplitude – The value of sin(x) , regardless of what form x is in, only has values from -1 to +1. So if I multiply the sine by any number, say 2.63, so that I now have y = 2.63 sin(x), then this results in values from 2.63 × (-1) = -2.63 to 2.63 × (+1) = 2.63. This make the amplitude of this new equation 2.63, that is, the number I multiply the sine by. So in general, the amplitude of y = sin(x) is A.
  2. Wavelength – Notice that the wavelength 365 is the denominator in
\[
\sin\left({\frac{360t}{365}}\right)
\]

The 360 in the numerator is the wavelength of the standard sine wave. The common symbol to represent wavelength is the Greek letter lambda, 𝝀, so in general, when you are taking the sine of something that looks like

\[
\sin\left({\frac{360t}{\mathit{\lambda}}}\right)
\]

the denominator, 𝝀, is the wavelength.

Now for those of you who have had exposure to this before, you may have expected to see 2𝜋t in the numerator instead of 360t. This would be the case if we were taking the sine of numbers expressed as radians. But this series of posts is doing everything with calculators in the degree mode. I will explain radians later in a different post.

3. Wave center – Notice that the center of the sine wave is at 12.165 which is the number added to the sine in the daylight length equation. The effect on a graph of adding a number to an equation is to raise or lower it – it does not change shape. So if you can graph and know the shape of y = something, then y = something + 10 will be the same shape, just shifted up 10 units. So adding 12.165 to the sine, doesn’t change its shape, it just changes where it is on the graph.

4. Time – The big change here is that we are no longer finding the sine of an angle. It may appear that we are now taking the sine of numbers in seconds, hours, or days – whatever the units of t are. However, the 360 in the numerator serves the purpose of making the number we are taking the sine of, unitless. That is, 360t/365 does not have any units – it is just a pure number.

Mathematicians/scientists long ago discovered that many periodic physical processes, have motions that follow a sine wave. In fact, when equations were formed that represented the forces on objects that were experiencing periodic motion, the sine of numbers involving time appeared when solving these equations.

And so it is with the length of the day throughout the year. The earth is rotating around the sun and this motion repeats, that is, is periodic. It is no surprise then that the graph of the day length is a sine wave.

I think we are now ready to model a mass on a spring. Let’s do that in my next post.

Graphs of Trig Equations, Part 2

In my last post, I showed that angles repeat every 360°. So an angle of 45° is the same as 45 + 360 = 405°. I also showed how angles can be negative if a reference line, like the positive x-axis is set up and angles created from that line going in the counter-clockwise direction are positive and going clockwise are negative. And I also showed that for angle 0°, sine 0° = 0 and sine 90° = 1. Please read my last post if needed.

Now without going through the development, it turns out that the sine has values that range from -1 to 1. Angles between 0° an 180° have positive sines and angles between 180° and 360° have negative sines. This repeats as one continues rotating around the x-axis.

Now we have already covered plotting equations so let’s plot the equation

y = sin x

where x is the angle:

So this is what a sine curve looks like. You can see that as you move along the x-axis, the curve moves up and down and repeats itself every 360°. The cosine curve is very similar but it is shifted to the left so that it begins at 1 when x = 0. So you see that the sine equation may prove useful when modelling something that repeats, like a mass on a spring bobbing up and down or a pendulum.

Now to prepare us for the modelling exercise which I will get to eventually, I want to define some characteristics of this sine curve.

First it has an amplitude. Amplitude is how high the curve goes above or below the center-line of the sine curve (or sine wave as it is frequently called). In this case, the center-line is the x-axis and the amplitude is 1 since the maximum extent of the curve is 1 unit above and below the center-line.

The sine wave has wave length. This is the distance between successive peaks (the highest points) or troughs (the lowest points). Lets look at the curve and measure the distance between any successive peaks. There is a peak at x = 90 and the next one is at x = 450. The distance between these two points on the x-axis is 450 – 90 = 360. This is what we expected as we know the sine curve repeats every 360° which is what wavelength means.

Associated with wavelength is something call frequency, but this will not make sense until I do a bit more development and include time in the mix. Stay tuned for the next post!

Graphs of Trig Equations, Part 1

Happy New Year and welcome to my first post of 2019!

My last post introduced the idea of modelling physical things with math equations. To do this from scratch, requires calculus but seeing the final result is very interesting. So in my last post, I modelled the simple physical event of a ball thrown into the air. Another common example when introducing modelling to students is a mass on a spring. But before I develop this, I want to show what the graphs of some trigonometric equations look like as they will be needed to describe any kind of motion that is cyclic, that is, repeats like a mass on a spring bobbing up and down.

So in a previous post, I defined what sin 𝜃 and cos 𝜃 are in terms of a right triangle. Given the below triangle

the sine and cosine of 𝜃 are defined as\[ \sin\mathit{\theta}\hspace{0.33em}{=}\hspace{0.33em}\frac{\mathrm{opp}}{\mathrm{hyp}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\cos\mathit{\theta}\hspace{0.33em}{=}\hspace{0.33em}\frac{\mathrm{adj}}{\mathrm{hyp}}\]

Let’s look at the sine for now. For very small angles, the opposite side will be small compared to the hypotenuse. Graphically, I think you can see that for an angle of 0°, there would be no opposite side so sin 0° = 0.

In the other extreme, as the angle gets close to 90°, the opposite side is close to the length of the hypotenuse, so the sine approaches 1. In fact,

sin 90° = 1.

Now angles are periodic in that they repeat every 360°. That is, an angle of 30° is also 30 + 360 = 390°. Another full circle of 360° can be added again to get an equivalent angle 390 + 360 = 750°. Angles can also be negative based on a convention of which direction you move to create the angle. Even with negative angles, multiple of 360° can be added or subtracted to get an equivalent angle whose sine will be the same. The below diagram shows these variations based on angles generated from the positive x-axis:

The angle in red is a positive angle, that is it is formed by going in the counter-clockwise direction from the x-axis. From that angle, you can go 1, 2, 3, etc complete circles to form the same angle. The angle in blue is a negative angle, that is it is formed by going clockwise from the x-axis. One can also go multiple complete circles around this angle to get the same angle. The point is that as you measure angles from 0, either in the positive or negative direction. you eventually repeat the same angles and these same angles will have the same sine value.

In my next post, I will plot the sine values against the angle values and show graphically what “periodic” means.

Graphing Equations

Well enough statistics, let’s return to some algebra topics. I’ve done a couple of posts on graphing and I would like to return to that.

So if you remember, a graph which we call the  cartesian coordinate system, is a way of plotting points in the form of (x, y) where x is the horizontal axis coordinate and y is the vertical coordinate. Below is a plot of several points on a graph:

But this is rather boring. Much more interesting is the graph of an equation which is a picture of all the x and y values that satisfy the equation. So if I have an equation y = x + 3, the plot of that equation is below with a few points labelled that show that they do indeed solve the equation, that is, make it true:

For example, the point (1, 4) solves this equation because when I substitute in x = 1 and y = 4, I get a true equation:

\[
{y}\hspace{0.33em}{=}\hspace{0.33em}{x}\hspace{0.33em}{+}\hspace{0.33em}{3}\hspace{0.33em}\Longrightarrow\hspace{0.33em}{4}\hspace{0.33em}{=}\hspace{0.33em}{1}\hspace{0.33em}{+}\hspace{0.33em}{3}\hspace{0.33em}\Longrightarrow\hspace{0.33em}{4}\hspace{0.33em}{=}\hspace{0.33em}{4}
\]

Plots of equations can be many shapes, but the concept is always the same: the plot is a picture of all the points that satisfy the equation. Let’s look at

\[
{y}\hspace{0.33em}{=}\hspace{0.33em}{x}^{2}\hspace{0.33em}{-}\hspace{0.33em}{4}
\]

If you just choose some x values and substitute them in and find the corresponding y values, you can get enough points plotted to show the approximate shape of the graph. For example, if x = 0, then y = -4. So (0, -4) is a point on the graph of this equation. If if x = 2, then y = 0, so (2, 0) is a point on the graph of this equation. Below is the graph of this equation with some points labelled:

Of course, there are an infinite number of points that satisfy this equation like (1.5, -1.75). The point is, this is a picture of all the points that satisfy the equation (within the plot borders of course as the graph goes up forever).

Graphs of equations are very useful in many areas of math, science, and engineering. In my next post, I’ll use a graph to show how it helps visualise a physical process like throwing a ball up in the air.

The Meaning of Graphs

In my last post, I showed how to plot points on a coordinate system. It is important to remember that a point such as (2, 1) means that for that point, x = 2 and y = 1. For the point (-5, -3), x = -5 and y = -3. The first number is always the x value and the second point is the y value. With that as a background, let’s talk about how to graph an equation.

Now we can solve an equation with one unknown like

\[{x}\hspace{0.33em}{+}\hspace{0.33em}{3}\hspace{0.33em}{=}\hspace{0.33em}{7}\]

Now past posts have talked about how to formally solve an equation like this but I think you can readily see that the solution to this equation is x = 4. That is, if you replace the x with 4, you get a true statement that 4 + 3 = 7. Now this is one equation with one unknown and there is only one solution. But what about y = x + 3 ? Here there are two unknowns, y and x. But you can come up with several solutions. If x = 4, then y = 7. If x = 5, then y = 8. If x = 1, then y = 4. If x = -2, then y = 1. You can see that there are many solutions to this one equation with two unknowns. In fact, there are an infinite number of solutions especially when you consider that fractional numbers are allowed as well. For example, if x = 2.67, then y = 5.67.

Now you see that the solutions are pairs of numbers: an x and a y. So we can think of a solution as a point on a graph. Since any point plotted on a coordinate system is of the form, (x, y), two of the solutions can be shown as (4, 7) and (-2, 1). If I plot these points and others that are solutions to the equation, it appears that all the points that make this equation true are on a line. in fact, they are and I’ve drawn the line over the points:

The main point here is that the graph of this equation is a picture of all the (xy) pairs that satisfy the equation. By looking at this, you can pick out other solutions like (3, 6) or (-3, 0). All the fractional points that satisfy this equation are also on the line. Since there are an infinite number of points that satisfy this equation, the graph is a solid line. Any graph, even curvy ones, are a picture of all the (xy) pairs that satisfy the equation that generated the graph.

The Plot Thickens

Doing maths looking at equations all day can sometimes get boring. Maths gets a lot more interesting when there are pictures. Most pictures in math involve graphs, so let’s start simple and begin with plotting points on a graph.

You are familiar with the number line:

You already know how to plot a point on this. But this is only a 1-dimensional plot. The most interesting thing you can plot on this is a horizontal line which represent all the numbers between the endpoints of the line. Wouldn’t it be nice if we could plot curves in 2-dimensions! Enter the cartesion coordinate system.

To the number line, let’s add a vertical number line intersecting the horizontal one at 0:

Each of these number lines are called an axis. The horizontal one is called the x-axis and the vertical one is called the y-axis. Now you can plot a point on any of these axes, but the strength of this system is that you can plot points anywhere on the surface that the coordinate system is on. But to plot a point in 2-dimensions, you need 2 numbers.

The typical way to indicate a point in maths is by using brackets. For example: (2, 1), (0, -7), (-4,3), (-1.5, -2.75). By convention, the first number is the x coordinate and the second number is the y coordinate, so the general point is (x, y). To plot a point, say (2, 1), you first go along the x axis 2 units to the right since 2 is positive, then go up 1 unit. That’s where the point (2, 1) is. I’ve plotted several other points below:

The point (0, 0) where the axes meet is called the origin. Note that positive x values are to the right of the origin and positive y values are above the origin. Negative values are to the left and below respectively.

Now this is cool but gets quickly boring just plotting points. The interesting things happen when we plot a set of points that satisfy an equation. I’ll get into that in my next post.